A genetic algorithm-based optimizing approach for project time-cost trade-off with uncertain measure
- Hua Ke^{1}Email author
DOI: 10.1186/2195-5468-2-8
© Ke; licensee Springer. 2014
Received: 12 February 2014
Accepted: 25 March 2014
Published: 24 April 2014
Abstract
Both the trade-off between the project cost and the project completion time and the indeterminacy of the environment are important issues for real-life project managers. In this paper, an uncertain time-cost trade-off problem, where activity cost functions are assumed to be linear and the objective function to be minimized is the project direct cost, is described based on uncertainty theory. Two uncertain time-cost trade-off models are built to satisfy different management requirements. To solve the proposed models, two equivalent crisp mathematical programming models are given, and genetic algorithm is introduced to search for quasi-optimal schedules. For future research, resource constraints or more types of indeterminacy can be included.
Keywords
Time-cost trade-off Uncertainty theory Uncertain measure Genetic algorithmIntroduction
For real-life projects, decision-makers should always consider the trade-offs among the performance goals for project scheduling and control, especially the trade-off between project completion time and project cost. The time-cost trade-off problem (TCTP) takes into account the project time-cost trade-off by crashing or prolonging project activity durations. In 1961, Kelly [1] first did research on the TCTP, which is one branch of the project scheduling problem. In the following years, many researches have been done on the deterministic TCTP [2, 3]. For solving the deterministic TCTP, the common analytical methods were linear programming and dynamic programming [4, 5]. Besides, some heuristic algorithms, such as genetic algorithm [6] and simulated annealing algorithm [7], were also applied. In recent years, other important factors for project scheduling, such as quality [8], have also been applied in research on TCTP.
As is well known, the real world is indeterminate. In real-life projects, the activity durations may be variational due to many external factors, such as the increase of productivity level and the change of weather. In recent years, many authors have considered the nondeterministic factors for describing the real-life project indeterminacy. In 1985, Wollmer [9] discussed a stochastic version of the deterministic linear TCTP. In 2000, Gutjahr et al. [10] designed a modified stochastic branch-and-bound approach and applied it to a specific stochastic discrete TCTP. Aghaie and Mokhtari [11] described an approach based on ant colony optimization method and Monte Carlo simulation technique for project crashing problem with exponentially distributed activity durations. Zahraie and Tavakolan [12] embedded two concepts of time-cost trade-off and resource leveling and allocation in a stochastic multiobjective optimization model, where fuzzy set theory was applied to represent different options for each activity. Ke et al. [13] built two models for stochastic TCTP with the philosophies of chance-constrained programming and dependent-chance programming. Mokhtari et al. [14] developed a hybrid optimization approach based on cutting plane method and Monte Carlo simulation for stochastic TCTP in PERT networks. Ke et al. [15] modeled stochastic project time-cost trade-offs with time-dependent activity durations.
For some projects in indeterminate environment, probability theory is no longer valid for describing activity durations for the lack of statistical data, since human beings usually overweigh unlikely events. For this case, the activity durations may be described by fuzzy variables. The first work on the fuzzy TCTP was done by Leu et al. [16]. Jin et al. [17] gave a genetic algorithm (GA)-based fully fuzzy optimal time-cost trade-off model, in which all parameters and variables were characterized by fuzzy numbers and an example in ship building scheduling was demonstrated. Eshtehardian et al. [18] established a multiobjective fuzzy time-cost model. Ghazanfari et al. [19, 20] applied possibilistic goal programming to the TCTP to determine optimal duration for each activity in the form of triangular fuzzy numbers. Ke et al. [21] built three fuzzy programming models for TCTP based on credibility theory. Chen and Tsai [22] constructed membership function of fuzzy minimum total crash cost based on Zadeh’s extension principle and transformed the time-cost trade-off problem to a pair of parametric mathematical programs.
When the indeterminacy does not behave either randomly or fuzzily, we need a new tool to deal with it. To describe indeterminacy which is neither randomness nor fuzziness, Liu founded uncertainty theory in 2007 [23] and refined it in 2010 [24]. Uncertainty theory is a branch of axiomatic mathematics for modeling human uncertainty. To the knowledge of the authors, no researchers considered time-cost trade-off problem in uncertain environment, which is not either stochastic or fuzzy. In this paper, we introduce uncertainty theory for modeling the TCTP in indeterminate environment. We propose two uncertain time-cost trade-off models according to some different decision-making criteria. As Huang and Ding [25] showed that using standard path algorithms (e.g., the well-known Dijkstra method) was not able to arrive at solutions for searching critical path of this problem, GA is applied in this paper. Some numerical experiments in which the activity durations are assumed to be uncertain variables with known uncertainty distributions are presented.
The remainder of the paper is organized as follows: Section ‘Preliminaries’ gives some concepts of uncertainty theory as preliminaries for modeling. Section ‘Problem description’ describes the TCTP with uncertain activity durations. In Section ‘Uncertain models of time-cost trade-off problem’, based on some decision-making criteria, two types of uncertain models are presented. The following section gives two numerical experiments to illustrate the proposed models. Finally, Section ‘Conclusions’ draws some concluding statements.
Preliminaries
To better describe and understand uncertain phenomena, Liu [23] proposed uncertainty theory, which has been applied to many fields such as uncertain calculus [26, 27], uncertain risk analysis [28], uncertain logic [29], uncertain finance [30], and uncertain differential equation [31]. Based on uncertainty theory, Liu [32] formulated uncertain programming for solving application problems with uncertain factors, which has been applied into some optimization problems, e.g., option pricing [33], facility location [34], inventory problem [35], transportation problem [36], uncertain graph [37], and portfolio selection [38].
In this section, we introduce some basic concepts which will be helpful for establishing some uncertain models for the TCTP. Let Γ be a nonempty set, and a σ-algebra over Γ. Each element Λ in is called an event.
Definition 1
where Λ_{ k } are arbitrarily chosen events from Ł_{ k } for k=1,2,…, respectively.
Based on the definition of uncertain measure, we can give the concept of an uncertain variable.
Definition 2.
is an event.
With the concept of uncertain variable, we can define the uncertainty distribution of an uncertain variable.
Definition 3.
for any real number x.
Liu [24] also defined the inverse function Φ^{−1} as the inverse uncertainty distribution of uncertain variable ξ. With inverse uncertainty distribution, Liu [24] gave the operational law of uncertain variables as follows.
Theorem 1
For giving out some decision-making criteria for managers, we introduce the following definition:
Definition 4.
provided that at least one of the above two integrals is finite.
Theorem 2.
Problem description
With the project progress, project managers always need to make a trade-off between the cost and the completion time. Sometimes managers may make a decision in order to finish the project sooner with project cost augmentation by accelerating the project schedule, which is also named as project crashing in project management. In other cases, motivated by reducing project cost, managers may be conscripted to sacrifice by prolonging the project completion time. Therefore, it is naturally desirable for managers to find a schedule to complete a project with the balance of the cost and the completion time.
Then, for the trade-off between the completion time and the cost, the goal is to decide the optimal vector x={x_{ i j }:(i, j)∈A} to meet different scheduling requirements.
Uncertain models of time-cost trade-off problem
In many researches on indeterminate decision systems, optimizing expected objective value is preferably considered to be the choice for decision-making and expected value model is the most employed model. However, for various practical requirements, other alternative decision-making criteria and optimization models are needed. In this paper, except the expected value model, one more optimization model for the uncertain TCTP is presented with the philosophy of dependent-chance programming.
Expected cost minimization model
where Ψ^{−1}(x, α) and Υ^{−1}(x, α) are determined by (2) and (4), respectively.
Chance maximization model
where Ψ^{−1}(x, α) and Υ^{−1}(x, β) are determined by (2) and (4), respectively.
Numerical experiments
By a simple example, Huang and Ding [25] demonstrated that the standard path algorithms (e.g., the well-known Dijkstra method) were not capable of finding the critical path for the random project scheduling problem, which is applicable to uncertain TCTP. Hence, for solving the above two uncertain time-cost trade-off models, GA is introduced to search for the quasi-optimal solutions.
Activity costs of project
Activity | Normal cost | Additional cost |
---|---|---|
(i,j) | c _{ ij } | d _{ ij } |
(1,2) | 170 | 200 |
(1,3) | 300 | 280 |
(1,4) | 65 | 70 |
(2,5) | 270 | 300 |
(3,5) | 135 | 150 |
(3,6) | 75 | 90 |
(3,7) | 150 | 100 |
(4,7) | 600 | 400 |
(5,8) | 85 | 100 |
(6,8) | 300 | 400 |
(6,9) | 95 | 90 |
(6,10) | 130 | 140 |
(7,10) | 200 | 240 |
(8,11) | 90 | 120 |
(9,11) | 200 | 180 |
(10,11) | 320 | 380 |
For the above model, we can easily employ GA to search for the quasi-optimal solution. We set the parameters in GA as the population size of one generation pop_size=50, the probability of mutation P_{m}=0.5, and the probability of crossover P_{c}=0.7. After a run of 8,000 generations, we obtain the quasi-optimal solution x^{∗}=(3,1,−4,3,3,−2,0,2,0,3,1,2,−2,0,0,−2), E[C(x^{∗}, ξ)]=43,482.5 and $\mathcal{\mathcal{M}}\left\{T\right({\mathit{x}}^{\ast},\mathit{\xi})\le 62\}=0.857$.
After a run of 6,000 generations with pop_size=50, P_{m}=0.6, and P_{c}=0.5, we obtain the quasi-optimal solution x^{∗}=(2,0,−4,3,2,0,1,2,3,3,0,3,0,1,3,−1) and $\mathcal{\mathcal{M}}\left\{C({\mathit{x}}^{\ast},\mathit{\xi})\le 46,700\right\}=0.874$.
Conclusions
For real-life project managers, both the trade-off between the cost and the completion time and the indeterminacy environment are considerable issues. In this paper, an uncertain TCTP was formulated with the objective of minimizing the cost with completion time limits based on uncertain measure. With some decision-making criteria, the expected cost minimization model and the chance maximization model were established to satisfy different practical managing requirements. Two equivalent crisp mathematical programming models were also given. To solve the models, GA was introduced.
The TCTP with uncertain activity durations can be regarded as the extension of the fuzzy TCTP. Furthermore, resource constraints or more types of indeterminacies can be included for future research. For real-life application, the models can be applied to many other project optimization problems.
Declarations
Acknowledgments
The work was partly supported by the National Natural Science Foundation of China (71371141, 71001080).
Authors’ Affiliations
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